The updated Directed Linear Arboricity Conjecture

Let $D$ be a directed graph, let $\Delta^-(D)$ and $\Delta^+(D)$ denote its maximum indegree and maximum outdegree, and let $\vec{la}(D)$ be the minimum number of colors needed to partition its arcs into directed linear forests. Let $K_n^*$ be the complete symmetric directed graph obtained by replacing every edge of $K_n$ by a pair of antiparallel arcs.

Directed Linear Arboricity Conjecture. For every directed graph $D$,

$$\vec{la}(D)\le\max\{\Delta^-(D),\Delta^+(D)\}+1,$$

except for $D=K_3^*$ or $K_5^*$, in which cases

$$\vec{la}(D)=\max\{\Delta^-(D),\Delta^+(D)\}+2.$$

The original conjecture was disproved by the two exceptional graphs, according to the source; this updated version is attributed to He, Li, Bai and Sun and is presented as unresolved.